Hardy spaces on Riemann surfaces under ramified coverings
Abstract
We extend the theory of indefinite Hardy spaces on finite bordered Riemann surfaces to the setting of ramified analytic coverings. Given a finite n-sheeted ramified covering F: S1→ S2 of finite bordered Riemann surfaces satisfying a spin-compatibility hypothesis, we construct (i) the direct image of a unitary flat vector bundle VxX1⊗ Del1 on the double X1 under F, taking full account of the ramification divisor RF and establishing the extension across the branch locus via a local analysis; (ii) a canonical matrix function G2 encoding the parahermitian structure on X2, together with the induced representation χ2 of piXX2p0; (iii) an explicit isometric isomorphism ϕF: H2,J1(p)(S1,VxS1⊗Del1) xrightarrow ∼ H2,J2(p)(S2,VxS2⊗Del2) between the associated Hardy-Kre\uın spaces, provided that h0(X1,VxX1⊗Del1)=0 and that the branch locus is disjoint from ∂ S2. We then develop the resulting operator theory in terms of vessels and Bezoutian operators. To each object in the category ℛℋ of finite bordered surfaces with unitary flat bundles we attach a triangular vessel whose input and output spaces are the Hardy-Kre\uın spaces on the two surfaces. The Bezoutian of the vessel is expressed as a finite-rank operator on ℋ2 whose kernel is built from bounded holomorphic point-evaluation functionals in ℋ2 evaluated at the interior ramification images F(rν)∈ S2, consistently with the boundary-transversality hypothesis ∂ S2∩ BF={∅}. We prove that the assignment (S,Vx,J)↦ H2,J(p)(S,Vx⊗Δ) extends to a covariant functor from ℛℋ (with ramified morphisms) to the category of Kre\uın spaces.
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BibTeX
@misc{zuevsky2026hardy,
title = {{Hardy spaces on Riemann surfaces under ramified coverings}},
author = {A. Zuevsky},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.22453},
eprint = {2606.22453},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.22453},
note = {CC0},
}
RIS
TY - UNPB
TI - Hardy spaces on Riemann surfaces under ramified coverings
AU - A. Zuevsky
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2606.22453
UR - https://arxiv.org/abs/2606.22453
AB - We extend the theory of indefinite Hardy spaces on finite bordered Riemann surfaces to the setting of ramified analytic coverings. Given a finite n-sheeted ramified covering F: S_1→ S_2 of finite bordered Riemann surfaces satisfying a spin-compatibility hypothesis, we construct (i) the direct image of a unitary flat vector bundle VxX1⊗ Del1 on the double X_1 under F, taking full account of the ramification divisor R_F and establishing the extension across the branch locus via a local analysis; (ii) a canonical matrix function G_2 encoding the parahermitian structure on X_2, together with the induced representation χ_2 of piXX_2p_0; (iii) an explicit isometric isomorphism ϕ_F: H^2,J_1(p)(S_1,VxS1⊗Del1) xrightarrow ∼ H^2,J_2(p)(S_2,VxS2⊗Del2) between the associated Hardy-Kre\uın spaces, provided that h^0(X_1,VxX1⊗Del1)=0 and that the branch locus is disjoint from ∂ S_2. We then develop the resulting operator theory in terms of vessels and Bezoutian operators. To each object in the category ℛℋ of finite bordered surfaces with unitary flat bundles we attach a triangular vessel whose input and output spaces are the Hardy-Kre\uın spaces on the two surfaces. The Bezoutian of the vessel is expressed as a finite-rank operator on ℋ_2 whose kernel is built from bounded holomorphic point-evaluation functionals in ℋ_2 evaluated at the interior ramification images F(r_ν)∈ S_2, consistently with the boundary-transversality hypothesis ∂ S_2∩ B_F={∅}. We prove that the assignment (S,Vx,J)↦ H^2,J(p)(S,Vx⊗Δ) extends to a covariant functor from ℛℋ (with ramified morphisms) to the category of Kre\uın spaces.
ER -